Historical data at a retail store (e.g. Wegmans, Giant, Safeway, Walmart, etc.) shows that the total average arrival rate of customers to checkout lanes (cashiers) at the store is 170 customers per hour during the peak hours. The arrivals can be modeled by a Poisson distribution. There are 17 cashiers working during peak hours. Each cashier can check out 11 customers/hr, on the average. The service times are exponentially distributed.

In Configuration 1 (separate-queue system), each cashier has a dedicated queue in front of them. This leads to 17 separate queues at the checkout area. In this configuration, assume that the arrivals of checkouts are equally distributed across the 17 queues: the average arrival rate to each queue = 170 customers per hour / 17 queues = 10 customers per hour per queue. The queueing environment in Configuration 1 includes 17 separate queues, each with 1 server.

In Configuration 2 (pooled-queue system), there is one serpentine queue which is served by all 17 cashiers. That is, all customers join the same queue for checkouts. Whenever a cashier becomes available, the customer in front of the queue gets served by that cashier. Note that the average arrival rate to the single checkout queue is 170 customers per hour in this configuration. The queueing environment in Configuration 2 includes one queue, and all customers get served by one of 17 servers, working in parallel.

A visual representation of Configuration 1 and Configuration 2 are provided in Figure-1 below

In order to determine the queueing performance, you will use the Excel spreadsheet provided in class. In Configuration-1, you can only analyze one queuing system (one cashier queue) at a time. Suppose we focus on the queueing system that is served by the first server. This queuing system has l1 =10 customers per hour, m=11 customers per hour, and S=1. For instance, the average number of customers waiting in one of the queues at any time would be Lq (use the given l1, m, S=1) on your spreadsheet. On the other hand, the total number of customers waiting in queues at the supermarket would be 17xLq, on the average. The average time a customer spends in the system is Ws (use the given l1, m, S=1 on your spreadsheet), regardless of which cashier queue the customer joins.

In Configuration-2, you can analyze the entire queueing system all at once: the queuing system has l=170 customers per hour, m=11 customers per hour, and S=17. The total number of customers waiting in queue at the supermarket at any point in time would be Lq, on the average (use the given l, m, S=17 on your spreadsheet). The average time a customer spends in the system is Ws (use the given l, m, S=17 on your spreadsheet).

a) (5 points) Fill in the table below by using the Queuing Models spreadsheet in the Excel file used in class:

Performance	Configuration 1	Configuration 2
Average wait time in the line (in minutes)*
Average wait time in the system (in minutes)*
Average server utilization rate*
Probability of delay (wait)*
Average number of customers in the line (per queue)*
Average number of customers in the line in total (considering all the queues)
Average number of customers in the system in total (considering all the queues and all the servers)

* You may compute this performance measure for only one cashier (i.e. one of the 17 identical queues) in Configuration-1. Notice that each cashier and their queue will have the same result on the average. Therefore, an average customer will have the same experience regardless of which queue (out of 17) they join in Configuration-1.

b) (1 point) Are the average server utilization rates equal in both configurations? Does the queue configuration affect the average utilization rates?

c) (2 points) Is the average time a customer spends in a queue in Configuration 1 the same as the average time a customer spends in the queue in Configuration 2? Explain clearly why or why not. NOTE: Read the articles in Question-2 to help you with this question.

c) (2 points) Suppose the supermarket trained their cashiers to serve their customers faster. The average service rate after the training will be 12 customers/hour. Repeat your analysis in part a and fill in the table below.

Performance	Configuration 1	Configuration 2
Average wait time in the line (in minutes)*
Average wait time in the system (in minutes)*
Average server utilization rate*
Probability of delay (wait)*
Average number of customers in the line (per queue)*
Average number of customers in the line in total (considering all the queues)
Average number of customers in the system in total (considering all the queues and all the servers)

d) (2 points) Based on your answers to parts (a) and (c), is a pooled-queue system (one serpentine queue leading to several servers), or a separate-queue system (one queue in front of each server) better for the customers? Explain clearly why this configuration performs better.

, 1 ui u 22 23 3 E CONFIGURATION 1: S separate, identical queues. Each queue has one server. hy+hy+z+...+hs=1 = = =... == /s Each queue is analyzed separately, as a single queue with one server. CONFIGURATION 2: One serpent queue. The queue has S identical servers. The system is analyzed as a single queue with S servers. Figure-1. A comparison of Configuration 1 to Configuration 2